Materials Fatigue Threshold From Jcost Fatigue Cost Zero At Yield

The yield baseline

Materials fatigue is the failure of metals and polymers under cyclic loading at stresses far below the static yield. The classical picture is the Wöhler curve, also called the S-N curve: a plot of stress amplitude against cycles to failure, which for many steels shows a fatigue limit, an endurance limit near half the yield stress, below which the material can endure indefinitely. Basquin's law describes the finite-life region as a power law, and the Coffin-Manson relation covers low-cycle fatigue. These empirical laws describe what happens, but not why the threshold sits where it does.

In Recognition Science, the framework models fatigue by a recognition cost, a number that measures how far a system is from a preferred baseline state. Here the baseline is the yield stress, the stress at which permanent deformation begins. The framework defines a per-cycle fatigue cost on the dimensionless stress ratio r, where r is the observed stress divided by the yield stress. The central theorem, fatigueCost_zero_at_yield, proves that this cost is exactly zero when r equals 1, meaning when the applied stress equals the yield stress. The cost is also symmetric under taking the reciprocal of the ratio and strictly positive for any ratio other than 1, so the yield point is the unique zero of the cost function.

The fatigue limit itself corresponds to a threshold value of this cost, the golden-section quantum, which the framework proves lies in the band between 0.11 and 0.13. Below this threshold, the framework defines the regime of infinite life, and at or above it, the regime of fatigue failure. The two regimes are proven mutually exclusive: a stress ratio cannot be both below and at or above the threshold. This places fatigue alongside plaque vulnerability, infarction, dysbiosis, ignition, accretion-disk transition, and magnetic reconnection as one universal quantum across pathology, combustion, plasma, and structural engineering.

What the theorem does not claim is that the cost function itself is derived from first principles in this file. The fatigue cost is defined as the J-cost from the framework's core, and the zero-at-yield property is inherited from that definition. The theorem establishes the mathematical structure, not the physical identification of the yield point as the baseline. The placement of the endurance limit in the band is a theorem about the cost function, not a measured value for any specific steel. The framework's library is a machine-checked collection of formal theorems, and this file contains zero unproven axioms, but the link between the formal cost and the physical fatigue data is a modeling choice, not a proved fact.

theorem fatigueCost_zero_at_yield : fatigueCost 1 = 0 := Cost.Jcost_unit0

theorem fatigueCost_reciprocal_symm {r : ℝ} (hr : 0 < r) :
    fatigueCost r = fatigueCost r⁻¹ := Cost.Jcost_symm hr

theorem fatigueCost_pos_off_yield {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < fatigueCost r := Cost.Jcost_pos_of_ne_one r hr hne

/-- Endurance threshold lies in the canonical band. -/
theorem endurance_threshold_band :
    0.11 < EnduranceThreshold ∧ EnduranceThreshold < 0.13 := by
  unfold EnduranceThreshold
  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  rw [Cost.Jcost_eq_sq hphi_ne]
  have h_lo : (1.61 : ℝ) < phi := Constants.phi_gt_onePointSixOne
  have h_hi : phi < (1.62 : ℝ) := Constants.phi_lt_onePointSixTwo
  have hpos : (0 : ℝ) < 2 * phi := by
    have : (0 : ℝ) < phi := Constants.phi_pos
    linarith
  refine ⟨?lo, ?hi⟩
  · rw [lt_div_iff₀ hpos]
    nlinarith [h_lo, h_hi]
  · rw [div_lt_iff₀ hpos]
    nlinarith [h_lo, h_hi]

/-- Infinite-life and fatigue-failing are mutually exclusive. -/
theorem regimes_exclusive {r : ℝ} :
    ¬ (IsInfiniteLife r ∧ IsFatigueFailing r) := by
  rintro ⟨h_lt, h_ge⟩
  exact (lt_irrefl _) (lt_of_lt_of_le h_lt h_ge)

What this page does not claim

The fatigue cost function is not derived from fatigue physics in this file; it is defined as the framework's J-cost. The endurance limit value is not a measured property of any specific steel; it is a theorem about the cost function's threshold. The physical link between the formal cost and real fatigue data is not proved; it is a modeling choice.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Materials/FatigueThresholdFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: